Elation generalized quadrangles with extra automorphisms and trivial spans

We show that the parameters of a finite elation generalized quadrangle for which the point spans containing the elation point (∞)(∞) are trivial, and which admit extra automorphisms fixing (∞)(∞) linewise, are powers of the same prime, so that the elation group is a pp-group for some prime pp.This observation has several strong corollaries, perhaps the most remarkable of which is that a thick elation quadrangle of order (s,s2)(s,s2) (for the natural number ss) that admits an extra automorphism of the aforementioned type which is an involution, is a translation generalized quadrangle—that is, the elation group is elementary abelian.